Does 'smutna' change based on the gender of the person you're speaking to?
Yes - mezczyzna jest smutny, kobieta jest smutna, dziecko jest smutne. (please excuse me for the missing accents)
I understand that "a sad child" would be "smutne dziecko", but if you were asking this question to a child would you use smutny/smutna?
You'd use the child's gender. If you were not sure about it, then you have a problem ;) 'smutne' is an option, I guess, but even a child could consider it weird.
It's perfectly fine, it's an accepted answer.
There isn't any real difference, really. But some people can consider 'czemu' a bit worse, a bit colloquial version, claiming that it's a Russicism.
Noun phrases need Instrumental, but standalone adjectives just take Nominative. Compare "Dlaczego jesteś smutną kobietą?" and "Dlaczego jesteś smutna?".